Introduction to Computational Fluid Dynamics

Transcription

1 Introducton to Computatonal Flud Dynamcs M. Zanub 1, T. Mahalakshm 2 1 (PG MATHS), Department of Mathematcs, St. Josephs College of Arts and Scence for Women-Hosur, Peryar Unversty 2 Assstance professor, Department of Mathematcs, St. Josephs College of Arts and Scence for Women-Hosur, Peryar Unversty Abstract: In CFD, we have flud problem. We use Naver-Stokes equaton to descrbe the physcal propertes of flud. The Naver-Stokes Equaton s analytcal. We want to solve ths equaton by computer, we have to translate t to the dscretzed form.at the end, and we can get our smulaton results. We can compare and analyze the smulaton results wth experments or the real problem. If the results are not suffcent to solve the problem, we have to repeat the process untl we fnd satsfed soluton. Keywords: Flud mechancs, Naver-Stokes equaton, numercal analyss, dscretzaton, grds, data structure and boundary condtons. I. INTRODUCTION To know what s CFD. Frstly, we have a flud problem. To solve ths problem, we should know the physcal propertes of flud by usng Flud Mechancs. Then we can use mathematcal equatons to descrbe these physcal propertes. Ths s Naver-Stokes Equaton. The Naver-Stokes Equaton s analytcal. Human can understand t and solve them on a pece of paper. But f we want to solve ths equaton by computer, we have to translate t to the dscretzed form. The translators are numercal dscretzaton methods, such as Fnte Dfference, Fnte Element methods. Consequently, we also need to dvde our whole problem doman nto many small parts because our dscretzaton s based on them. Then, we can wrte programs to solve them. The typcal languages are FORTRAN and C. runnng the programs on workstaton or supercomputer. At the end, we can get our smulaton results. We can compare and analyze the smulaton results wth experments or the real problem. If the results are not suffcent to solve the problem, we have to repeat the process untl fnd satsfed soluton. Ths s CFD. II. OBJECTIVES The obectve of CFD s to model the contnuous fluds wth Partal Dfferental Equatons (PDEs) and dscretze PDEs nto an algebra problem (Taylor seres), solve t, valdate t and acheve smulaton based desgn usng computer. IJRASET (UGC Approved Journal): All Rghts are Reserved 141

2 Here s a comparson table of Smulaton and Experment. From ths table, we can see that Smulaton s much cheaper that experment because we do not need to buy the expensve experment equpment s. We can get the results n short tme wth CFD. We can do smulaton for any scale of problem. For example, from the small bubbles to weather of earth. But experment can only work for the small or mddle sze obect. From Smulaton, we can get any nformaton we need. But n experment, we can only obtan data from the measured pont. Smulaton s very easy to repeat, we only need to run the program agan. But experment s not so easy, especally the combuston, explosons. They are unrepeatable. Some experments are very dangerous, for example, polluton and radaton. But f we use Smulaton, t s very safe. A. Physcs of flud The frst thng we should make clear s what flud s. Flud s lqud and gas, for example, water and ar. Flud has some mportant propertes, for example, Pressure, velocty, temperature and mass. Here I want to emphass two propertes. The frst one s densty. In flud mechancs, f densty s constant, we call the flud s ncompressble flud. Sometmes, f the change of the densty s very small, we can also treat the flud as ncompressble flud. For example, water. If densty s varable, we call the flud s compressble. For example, ar s compressble flud. Later on we wll see the mathematcal equaton for ncompressble flud s much smpler than compressble flud. Another mportant property s vscosty. Vscosty s an nternal property of a flud that offers resstance to flow. For example, to str water s much easer than str honey because the vscosty of water s much small than honey. Ths table shows the densty and vscosty of ar, water and honey. III. CONSERVATION LAW: Ths pcture shows the prncple of conservaton law. The change of the mass s equal to the mass flow n mnus mass flow out. If the mass flow n s equal to mass flow out, then the change of mass s zero. Actually, the conservaton law s not only for mass, but also for momentum and energy. A. Naver-Stokes Equaton Usng the Conservaton law, we can derve the mathematcal equatons for flud. These equatons are Naver-Stokes equatons. The Naver-Stokes equatons are a set of nonlnear partal dfferental equatons that descrbes the flow of heat. It assumes the forms of contnuty equaton, equaton of moton and conservaton of energy. The frst one s Contnuty Equaton, t comes from Mass IJRASET (UGC Approved Journal): All Rghts are Reserved 142

3 Conservaton. Dp/Dt s the change of the mass, ρdu/dx s convectve term, whch means the mass flux. Ths s for compressble flud because the densty can change wth tme. If densty s constant, whch means Dp/dt s zero. Ths s for ncompressble flud. t I II P g x V III IV \ x 2 3 xk k If we apply Momentum Conservaton, we can get momentum equaton. The frst term s local momentum change wth tme. The second term s convectve term, or we can say momentum flux. The thrd term s momentum change due to surface force. We can mage that the pressure s actve at the surface of obect, and the surface force can change the momentum of obect. The fourth term s momentum exchange wth molecular moton. The momentum of the obect can transfer to momentum of molecules. The last term s momentum change due to Mass Force. For example, the gravtatonal force, acceleraton force. The prevous momentum equaton s for compressble flud. For ncompressble flow, the contnuty equaton s du/dx=0. Whch means the duk/dxk s zero, so the later term s zero. du/dx=0, So the momentum equaton for ncompressble flud can be wrtten as ths formula. 2 T T T c c P 2 t I II III IV V If we use energy conservaton law, we can get energy equaton. The frst term s local energy change wth tme. The second term s convectve term. The thrd s heat flux. Fourth s the work done by pressure. The last one s the transfer of mechancal energy nto heat. We have get Naver-Stokes equatons. But these equatons are analytcal equatons. Human can understand and solve, but computer cannot. So we need to translate them to the forms whch computer can understand. Ths process s dscretzaton. The typcal dscretzaton methods are Fnte Dfference, Fnte Element and Fnte volume methods. B. Grd Generaton From fnte volume method, we know that we need to dvde the whole problem doman nto many small domans, and then ntegrate at these small domans. Ths s Grd Generaton. We have 3 methods to generate the grds. The smplest one s structured IJRASET (UGC Approved Journal): All Rghts are Reserved 143

4 grd. In ths type of grds, all nodes have the same number of elements around t. We can descrbe and store them easly. But ths type of grd s only for the smple doman. If we have a complex doman, we can use unstructured grd. Generally, unstructured grd s sutable for all geometres; t s very popular n CFD. The dsadvantage s that because the data structure s rregular, t s more dffcult to descrbe and store them. Block structure grd s a compromsng of structured and unstructured grd. The dea s, frstly, dvde the doman nto several blocks, and then use dfferent structured grds n dfferent blocks. [3] C. Boundary Condtons To solve the equaton system, we also need boundary condtons. The typcal boundary condtons n CFD are No-slp boundary condton, Axsymmetrc boundary condton, Inlet, outlet boundary condton and Perodc boundary condton. For example, there s a ppe, the flow comes n from the west, comes out from the east sde. So we can use nlet at the west sde, whch means we can set the velocty manually. At the west sde, we use outlet boundary condton to keep all the propertes constant at x drecton, whch means the gradent s zero. At the wall of ppe, we can set the velocty s zero, ths s no-slp boundary condton. At the center of ppe, we can use Axsymmetrc boundary condton. Perodc boundary condtons are a set of boundary condtons whch are often chosen for approxmatng a large (nfnte) system by usng a small part called a unt cell. D. Solvers Drect: Cramer s rule, Gauss elmnaton, LU (lower upper) decomposton. Iteratve: Jacob method, Gauss-Sedel method, Successve over relaxaton method. E. Numercal Parameters Under relaxaton factor, convergence lmt. Montor resduals (change of results between teratons). Number of teratons for steady flow or number of tme steps for unsteady flow. Sngle/double precsons. Multgrd. Parallelzaton. F. Applcatons 1) Industral Applcatons: CFD s used n wde varety of dscplnes and ndustres, ncludng aerospace, automotve, power generaton, chemcal manufacturng, polymer processng, petroleum exploraton, pulp and paper operaton, medcal research, meteorology, and astrophyscs. Example: Analyss of Arplane. CFD allows one to smulate the reactor wthout makng any assumptons about the macroscopc flow pattern and thus to desgn the vessel properly the frst tme. Chemcal ndustry s IJRASET (UGC Approved Journal): All Rghts are Reserved 144

5 another mportant CFD applcaton feld. polymerzaton reactor vessel. It helps n the predcton of flow separaton and tme effect of resstance n IV. CONCLUSION CFD s a method to numercally calculate heat transfer and flud flow. Currently, ts man applcaton s as an engneerng method, to provde data that s complementary to theoretcal and expermental data. Ths s manly the doman of commercally avalable codes and n-house codes at large companes. CFD can also be used for purely scentfc studes, e.g. nto the fundamentals of turbulence. Ths s more common n academc nsttutons and government research laboratores. Codes are usually developed to specfcally study a certan problem. REFERENCES [1] Anderson J.D Computatonal Flud Dynamcs [2] Computatonal Flud Dynamcs and Heat Transfer by Anderson D.A., Tenehll J.C. and Pletcher R.H [3] An Introducton to Computatonal Flud Dynamcs: The Fnte Volume Method H.Versteeg [4] Computatonal Flud Dynamcs: A Practcal Approach by Tu [5] CFD-WK, the free CFD reference [6] Computatonal Flud Dynamcs-Hoffmann-cted by 1129(artcle) [7] The Fundamentals of Computatonal Flud Dynamcs-Hrsch-cted by 6243(artcle). Hoffmann, Klaus A, and Chang, Steve.T Computatonal flud dynamcs for engneer s vol. I and vol. II [8] Raesh Bhaskaran, Lance Collns Introducton to CFD Bascs [9] accessed on 11/10/06. Adapted from notes by: Tao Xng and Fred Stern, The Unversty of Iowa. IJRASET (UGC Approved Journal): All Rghts are Reserved 145